Partial Differential Equations

An Introduction to Analytical and Numerical Methods
1. Auflage 2023
ISBN: 978-3-031-13381-7
Verlag: Springer International Publishing

Buch, Englisch, Band 294, 452 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 715 g

Reihe: Graduate Texts in Mathematics

This textbook introduces the study of partial differential equations using both analytical and numerical methods. By intertwining the two complementary approaches, the authors create an ideal foundation for further study. Motivating examples from the physical sciences, engineering, and economics complete this integrated approach.

A showcase of models begins the book, demonstrating how PDEs arise in practical problems that involve heat, vibration, fluid flow, and financial markets. Several important characterizing properties are used to classify mathematical similarities, then elementary methods are used to solve examples of hyperbolic, elliptic, and parabolic equations. From here, an accessible introduction to Hilbert spaces and the spectral theorem lay the foundation for advanced methods. Sobolev spaces are presented first in dimension one, before being extended to arbitrary dimension for the study of elliptic equations. An extensive chapter on numerical methods focuses onfinite difference and finite element methods. Computer-aided calculation with Maple™ completes the book. Throughout, three fundamental examples are studied with different tools: Poisson’s equation, the heat equation, and the wave equation on Euclidean domains. The Black–Scholes equation from mathematical finance is one of several opportunities for extension.

Partial Differential Equations offers an innovative introduction for students new to the area. Analytical and numerical tools combine with modeling to form a versatile toolbox for further study in pure or applied mathematics. Illuminating illustrations and engaging exercises accompany the text throughout. Courses in real analysis and linear algebra at the upper-undergraduate level are assumed.

Zielgruppe

Graduate

Autoren/Hrsg.

Arendt, Wolfgang

Urban, Karsten

Weitere Mitwirkende

Kennedy, James B.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik

Inhaltsverzeichnis

1 Modeling, or where do differential equations come from.- 2 Classification and characteristics.- 3 Elementary methods.- 4 Hilbert spaces.- 5 Sobolev spaces and boundary value problems in dimension one.- 6 Hilbert space methods for elliptic equations.- 7 Neumann and Robin boundary conditions.- 8 Spectral decomposition and evolution equations.- 9 Numerical methods.- 10 Maple®, or why computers can sometimes help.- Appendix.

Über Autor(innen)

Wolfgang Arendt is Senior Professor of Analysis at Ulm University. His research areas are functional analysis and partial differential equations.

Karsten Urban is Professor of Numerical Mathematics at Ulm University. His research interests include numerical methods for partial differential equations, especially with concrete applications in science and technology.